An algebraic theory for multigrid methods for variational problems
SIAM Journal on Numerical Analysis
On the algebraic multigrid method
Journal of Computational Physics
Element-Free AMGe: General Algorithms for Computing Interpolation Weights in AMG
SIAM Journal on Scientific Computing
AMGE Based on Element Agglomeration
SIAM Journal on Scientific Computing
Algebraic Multigrid Based on Element Interpolation (AMGe)
SIAM Journal on Scientific Computing
Iterative Methods for Sparse Linear Systems
Iterative Methods for Sparse Linear Systems
An Algebraic Multigrid Method for Linear Elasticity
SIAM Journal on Scientific Computing
Multiple Vector Preserving Interpolation Mappings in Algebraic Multigrid
SIAM Journal on Matrix Analysis and Applications
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Based on the geometric grid information as geometric coordinates, an algebraic multigrid (AMG) method with the interpolation reproducing the rigid body modes (namely the kernel elements of semi-definite operator arising from linear elasticity) is constructed, and such method is applied to the linear elasticity problems with a traction free boundary condition and crystal problems with free boundary conditions as well. The results of various numerical experiments in two dimensions are presented. It is shown from the numerical results that the constructed AMG method is robust and efficient for such semi-definite problems, and the convergence is uniformly bounded away from one independent of the problem size. Furthermore, the AMG method proposed in this paper has better convergence rate than the commonly used AMG methods. Simultaneously, an AMG method that can preserve the quotient space, which means that if the exact solution of original problem belongs to the quotient space of discrete operator considered, then the numerical solution of AMG method is convergent in the same quotient space, is obtained using the technique of orthogonal decomposition.